Odd question.
Germany had occupied all of the Atlantic coastlines of France, as well as Paris, to my understanding mostly for unhindered access to the infrastructure and naval bases, as well as other reasons, so cities like Brest, Paris, and Borduex, but the Mediterranean cities (aside from Nice) were left unoccupied, so why weren't one of those cities chosen to be the capitol?
But why wasn't a larger or arguably more important city chosen to be the capitol, such as Marseille, Monpellier, or Lyon designated as the capitol?
Once again I'm aware it's a really odd question but Vichy kind of seems like an arbitrary choice, and I'd like to know if this was the case or if there was a strategic decision behind it
I'll give you a quick answer, which is basically a translation of this article : https://www.futura-sciences.com/sciences/questions-reponses/histoire-t-on-installe-gouvernement-vichy-5567/
First thoughts for the new capital were places like Versailles, Bourges and Fontainebleau, that had been centers of power in the past, but all of these were in occupied territory.
The capital had to be in "free territory". Moulins and Clermont Ferrand were considered because of that, as they were pretty close to the division line, but they were judged too blue collar and not comfortable enough by the people in power.
Raphael Alibert and Paul Beaudoin suggested Vichy as it had good train connections to Paris and a great hotel industry. That second factor was decisive for the choice.
Lyon, Montpellier and most bigger towns further South did not have direct train connection to Paris (to be honest, that only came in the past 20 years) and we're further from a central position. They also didn't have as much in terms of luxury hotels, as fancy hotels and casinos were basically found in thermal towns (not sure what the proper word is in English, but we are talking about towns with hot springs and such that were used by rich people to get better / relax thanks to the healing properties of these hot springs).